Vol 80, No 2 (2025)

Many concepts in differential geometry (for example, Riemannian metric, length, shortest arc, volume), defined originally for manifolds, are easily transferred to simplicial polyhedra. This extends significantly the scope of application of such concepts familiar in the framework of manifolds as systolic volume and volume entropy. Many properties of these invariants depend only on the fundamental group of the polyhedron. Using the process of minimisation, it becomes possible to transfer these invariants directly to finitely presented groups. In studying the systolic area and volume entropy of groups, new combinatorial invariants arise, which are of independent interest. The paper is an introduction to this area at the intersection of geometry, topology, and group theory.Bibliography: 65 titles.

This survey sums up a cycle of papers devoted to the construction of finite-dimensional moduli spaces points in which are certain special Lagrangian submanifolds of compact complex simply connected algebraic varieties. The starting point for this construction was the idea, due to A. Tyurin, to treat Largrangian submanifolds (or equivalence classes of such submanifolds) as mirror counterparts of stable vector bundles. Our constructions are based on the programme of abelian Lagrangian algebraic geometry developed by A. Tyurin and Gorodentsev 25 years ago. Since this programme was in its turn based on the Bohr–Sommerfeld Lagrangian geometry known in geometric quantization, we call our construction special Bohr–Sommerfeld geometry. The definitions arising in the course of work turn out to be closely connected with the theory of Weinstein domains, Eliashberg's conjectures, and many other concepts in symplectic geometry. The core conjecture that arose in our work and is confirmed by the available examples states that each moduli space of this type is in its turn an algebraic variety.Bibliography: 13 titles.